- Email: [email protected]
Entropy estimation of the Hénon attractor
Chaos, Solitons & Fractals 45 (2012) 805–809
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Entropy estimation of the Hénon attractor Chihiro Matsuoka a,⇑, Koichi Hiraide b a b
Department of Physics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japan Department of Mathematics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japan
a r t i c l e
i n f o
Article history: Received 8 June 2011 Accepted 18 February 2012 Available online 31 March 2012
a b s t r a c t The topological entropy of the Hénon attractor is estimated using a function that describes the stable and unstable manifolds of the Hénon map. This function provides an accurate estimate of the length of curves in the attractor. The estimation method presented here can be applied to cases in which the invariant set is not hyperbolic. From the result of the length calculation, we have estimated the topological entropy h as h 0.49703 for the original parameters a = 1.4 and b = 0.3 adopted by Hénon. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The Hénon map [1] is a model that depicts fluid turbulence from the viewpoint of a dynamical system [2]; in this map, the topological entropy of the attractor is the quantity that measures the complexity of fully developed turbulence [3,4]. Generally, when the topological entropy is known, we can use it to calculate thermodynamical quantities such as the Gibbs measure or the pressure of a system. Owing to the significance of entropy in thermodynamics of dissipative systems or the probability theory, several studies have calculated the topological entropy of the Hénon attractor by various methods. In the case of a two-dimensional system, the topological entropy can be defined as supremum of the growth ratio of lengths of curves in invariant sets (usually, the unstable manifold) in a system [5–7]. Mori and Fujisaka calculated the topological entropy h and the Housdorff dimension D of the Hénon map by the method of map iteration and obtained the values of h 0.42 and D 1.26. Biham and Wenzel introduced a Hamiltonian associated with the Hénon map and estimated as h 0.67 (note that their definition of topological entropy is slightly different from the one presented here) by counting the number of periodic points up to period p = 28 [8]. Their numerical technique can also be applied ⇑ Corresponding author. E-mail address: [email protected] (C. Matsuoka). 0960-0779/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2012.02.013
to depict Julia sets in complex dynamical systems [9]. Galias and Zgliczyn´ski [10] and Galias [11] calculated the topological entropy by counting periodic points up to period p = 30, in which they estimated the value h 0.46486. Gelfert and Kantz adopted the algorithm of Biham and Wenzel and use it to calculate the topological entropy and pressure for various values of dynamical parameters a and b [see (1)] by counting periodic points up to period p = 31, by which they estimated as h 0.44–0.46 for the Hénon map [12]. Newhouse et al. performed numerical calculations with rigorous error estimates and proved the result of h P 0.46469 [13]. When the invariant set is hyperbolic, the topological entropy is usually given by counting the number of periodic points [14]; however, it is very difficult to obtain an accurate value of topological entropy in cases where the invariant set is not hyperbolic or the unstable manifold is not a true attractor. Although a rigorous mathematical proof does not exist, the invariant set of the Hénon map is considered to be non-hyperbolic [15]; therefore, the method of calculating periodic points [8,12] does not always give an accurate entropy value. The method of counting parts of the stable and unstable manifolds created by the map iteration presented by Newhouse et al. [13] gives an accurate value; however, complicated numerical calculations are required for this method. Matsuoka and Hiraide found a function that describes the stable and unstable manifolds of the Hénon map and depicted these manifolds using this function [16]. This function provides the
806
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
accurate length of the unstable manifold; therefore, we can calculate the topological entropy by using this function regardless of whether or not the invariant set is hyperbolic, i.e., whether or not the unstable manifold is an true attractor. In this paper, we report estimation of the topological entropy of the Hénon map by adopting this function. We mention that this estimation method is applicable to various values of dynamical parameters a and b [see Definition (1)]. This paper is organized as follows. In Section 2, we present basic facts and notations adopted in this paper, and briefly review the asymptotic expansion form to describe the stable and unstable manifolds of the Hénon map. In Section 3, we provide the definition of the topological entropy and perform the numerical estimate of that using the asymptotic expansion presented in the previous section. A figure of the stable and unstable manifolds of the Hénon map represented by this asymptotic expansion is also presented in this section. Section 4 is devoted to conclusion. 2. Asymptotic expansion of stable and unstable manifolds of Hénon map The Hénon map [1] is an invertible nonlinear transformation defined by a polynomial map f : C2 ! C2
f ðx; yÞ ¼ ð1 þ y ax2 ; bxÞ;
ð1Þ
where a and b are complex parameters; and we set a = 1.4 and b = 0.3 when estimating the topological entropy. When a fixed point P = (xf, yf) is a saddle point, two eigenvalues a1 and a2 at P are given by the solutions to the quadratic equation a2 + 2axfa b = 0, where 0 < ja1j < 1 and ja2j > 1. We define the stable and unstable manifolds at P by
N log N N ðNÞ K N ja1 j bN;N1 6 N!
for the stable manifold and as N log N 0N ðNÞ K N ja2 j bN;N1 6 N!
for the unstable manifold, where KN > 0 and K 0N > 0 are constants. These estimates guarantee the suppression of divergence of the factor eNf1 t ðN 1Þ! for Re(f1t) > 0 in (3). Here, we mention that the asymptotic expansion (3) is not a formal one; that is, this expansion is not the divergent series (for details, see Ref. [16]). In order to depict the stable and unstable manifolds, we introduce a large positive integer n such that
xðt nÞ
1 P
ðNÞ
eNf1 ðtnÞ
N¼1
bN;N1 ðN 1Þ! ðtÞN
;
ð4Þ
where the upper (lower) sign in (±) corresponds to the stable (unstable) manifold. We denote the integer part of jRe(t) nj as M; then the integer M corresponds to the map iteration number (minus 1), i.e., f(M+1)[x(t)] = x(t n). When Re(t) > 0 and Re(t) n < 0, the curve (x, y) = [x(t n), bx(t n 1)] describes the stable manifold; when Re(t) > 0 and Re(t) n > 0, the curve [x(t n), bx(t n 1)] tends to the fixed point P owing to the relation limM?1fM[x(t)] ? P. When Re(t) < 0 and Re(t) + n > 0, the curve (x, y) = [x(t + n), bx(t + n 1)] describes the unstable manifold; when Re(t) < 0 and Re(t) + n < 0, the curve [x(t + n), bx(t + n 1)] tends to the fixed point P owing to the relation limM?1fM[x(t)] ? P. In order to calculate the length of the unstable manifold, we adopt the expansion (4). 3. Topological entropy of the Hénon attractor
W s ðPÞ ¼ fQ 2 C2 jf n ðQ Þ ! P as n ! 1g and
3.1. Definition and numerical estimate of topological entropy
W u ðPÞ ¼ fQ 2 C2 jf n ðQ Þ ! P as n ! 1g;
Following Yomdin, and Newhouse et al. [5–7], we adopt the topological entropy h as
respectively. After shifting the fixed point P to the origin by x ? x + xf and y ? y + yf, we introduce a parameter t 2 C that parameterizes the stable and unstable manifolds as f[x(t), y(t)] = [x(t + 1), y(t + 1)]. Then, the Hénon map (1) yields the difference equation
xðt þ 1Þ kxðtÞ bxðt 1Þ ¼ afxðtÞg2 ;
ð2Þ
associated with y(t) = bx(t 1), where k = 2axf. Using the Borel–Laplace transform, Matsuoka and Hiraide found solution x(t) to (2), whose asymptotic expansion is given as [16]
xðtÞ
1 P N¼1
ðNÞ
Nf1 t
e
bN;N1 ðN 1Þ! ðtÞN
;
ð3Þ
where the + and signs correspond to the stable and unstable manifolds, respectively, and f1 is given by f1 = logja1j for the stable manifold and f1 = logja2j + pi ðNÞ for the unstable manifold. The coefficient bN;N1 is a constant given by a recurrence formula (Theorem 2 in [16]) and is estimated as
X Lkþ1 1 M1 ln ; M!1 M Lk k¼0
h ¼ lim
ð5Þ
where Lk is the length of the kth piece (the length obtained by the (k + 1)th iteration of the map f) on the Hénon attractor. Here, the length L of a curve is approximately given by
L¼
K1 P
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½xðtmþ1 Þ xðt m Þ2 þ ½yðtmþ1 Þ yðt m Þ2 ;
ð6Þ
m¼0
where the set [x(tm), y(tm)] denotes the coordinate on the (x, y) plane that is parameterized by the parameter tm, i.e., the discretized t. If the difference Mtm tm+1 tm is sufficiently small, L in (6) can be expected to well approximate the true length of the curve Ltrue ¼ ffi R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdx=dtÞ2 þ ðdy=dtÞ2 dt. We adopt formula (6) for calculating the partial length Lk (k = 0, 1, 2, . . .) numerically. Here, we restrict ourselves to the curves in the unstable manifold.
807
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
Since we cannot take M ? 1 in (5) numerically, we define the partial entropy hl,M for numerical calculations as follows:
hl;M ¼
1 X 1 M Lkþ1 ln : M l k¼l Lk
ð7Þ
As is seen easily, h is given by h = liml?0 limM?1hl,M. We use the asymptotic expansion (4) to calculate the length of the Hénon attractor. The algorithm for deriving the length Lk in (7) is as follows. First, we calculate the ðNÞ coefficient bN;N1 in (4) by using the recurrence formula in Theorem 2 in Ref. [16]. In order to avoid a round-off erðNÞ ror, the calculation of bN;N1 is performed with 300 digits. The upper limit of the summation in (4) is taken as N = 600. Then, we calculate x(t + n) (the unstable case) in ðNÞ (4) by using this bN;N1 , where f1 = logja2j + pi 0.6543 + pi for a = 1.4, b = 0.3, and a2 1.9237. We set n = 20,000 here. For this n, we consider the region 20,000 6 Re(t) 6 19,978, so the map iteration number (minus 1) M; i.e., the integer part of jRe(t) + nj, is in the range 0 6 M 6 22. As can be seen from the value of f1, the value of x(t + n) with respect to real t is not real for the unstable manifold; therefore, we must seek points such that the imaginary part of (x, y) is close to zero on C2 in order to obtain real (x, y). To do this, we divide the complex t-plane into pieces and search the points at which the distance from the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi real axis satisfies the condition sðtÞ xi ðtÞ2 þ yi ðtÞ2 6 ð 1Þ, where xi(t) = Im[x(t + n)] and yi(t) = Im[bx (t + n 1)]. We select = 103 in this study. The points that satisfy this condition correspond to tm in (6). Upon setting t = tr + i ti, we found that tm appears on a line with slope ti/ tr 4.8 [16]. The sets [x(tm), y(tm)] obtained by this algorithm accurately depict the unstable manifold of the Hénon map (see Fig. 3). For a small map iteration number M, we can find points t = tm that satisfy s(t) 6 103 up to the neighborhood of the next integer M + 1, when the starting point of the detection is taken to be as M. For example, we can find such points approximately up to M + 0.9 for M = 0 and 1 (tr = 20,000 and 19,999, respectively), M + 0.8 for M = 2 and 3, and M + 0.7 for M = 4 and 5, starting from M. However, these points gradually decrease in number owing to the roundoff error, and finally, we few such points are found for M P 23. We find s(t) 6 103 points up to M + 0.1 for M = 22 (tr = 19,978); therefore, taking this value into account, we select the interval M 6 tr 6 M + 0.1 in tr (ti = 4.8tr) for all M. Dividing the interval 0.1 in one integer M into equally spaced grid points with a distance of Mtr (Mti = 4.8Mtr), we calculate the length of curves for all M (0 6 M 6 22). Here, we set Mtr = 1.0 104 for 0 6 M 6 7, Mtr = 5.0 105 for 8 6 M 6 17, and Mtr = 1.0 105 for 18 6 M 6 22; that is, we consider 1000, 5000, and 10,000 grid points for 0 6 M 6 7, 8 6 M 6 17, and 18 6 M 6 22, respectively. The calculations for x(t + n) = f(M+1)[(x(t)] are performed with 32, 64, 128, and 300 digits for 0 6 M 6 7, 8 6 M 6 17, 18 6 M 6 21, and M = 22, respectively (see Fig. 3). By using the above algorithm, we find [x(tm), y(tm)] that would provide the unstable manifold and calculate the
length LM (0 6 M 6 22) from formula (6). These lengths for a = 1.4 and b = 0.3 are listed in Table 1. The lengths LM for 0 6 M 6 11 are very small and the points [x(tm), y(tm)] for these M values stay in the neighborhood of the fixed point. Indeed, almost all parts of the Hénon attractor are depicted by the points [x(tm),y(tm)] for 12 6 M 6 22. The partial topological entropy hl,M in (7) for 0 6 M 6 11 varies widely and we cannot obtain an accurate value for it. Therefore, we do not adopt these LM (0 6 M 6 11) values for the topological entropy calculation. Fig. 1 shows the partial topological entropy h12,M, for which l = 12 in (7). After varying at relatively small M values, the partial topological entropy h12,M approaches a certain value. The h12,M values for 19 6 M 6 22 are estimated as h12,19 0.53874, h12,20 0.50556, h12,21 0.49853, and h12,22 0.49703. We regard this last h12,22 value as the asymptotic value of the topological entropy h in the approximation up to M = 22, i.e.,
h 0:49703:
ð8Þ
Our result supports the estimate of h P 0.46469 by Newhouse et al. [13]. We remark that Fig. 1 mathematically
Table 1 Length of curves in the unstable manifold. The first and third columns show the iteration number M, and the second and fourth columns show the partial length of the attractor, LM, at the iteration number M. M
LM
M
LM
0 1 2 3 4 5 6 7 8 9 10 11
7.1843 105 1.3821 104 2.6583 104 5.1151 104 9.8341 104 1.8938 103 3.6353 103 7.0211 103 1.3402 102 2.6162 102 4.8879 102 9.9124 102
12 13 14 15 16 17 18 19 20 21 22
0.17008 0.39014 0.44008 1.31912 1.65998 2.68724 4.77124 7.38664 9.70860 15.1093 24.5045
Fig. 1. Partial topological entropy h12,M versus map iteration number M, where the white circles show the calculation points of h12,M.
808
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
converges with M ? 1; however, the result up to M = 22 is best possible in actual calculations at present. The values obtained by Newhouse et al. and in the present study are larger than the value calculated by counting the periodic points [12]. Katok [17] proved a theorem that when the system is two-dimensional and the topological entropy h > 0, there exist hyperbolic subdynamical systems having entropy h infinitely close to h. This h is possible to calculate by counting the number of periodic points. The difference between the result presented by Gelfert and Kantz (h 0.44–0.46) and ours (h 0.49703) seems to suggest that there exist numerous periodic points with longer period in the Hénon attractor that cannot count within the period p = 31 considered in Ref. [12]. We show the topological entropy for various values of a in the neighborhood of a = 1.4 in Fig. 2, where the value of b is fixed at b = 0.3. The accuracy of the calculation is identical with that of Fig. 1. We find the devil’s staircase-like structure for the topological entropy, which resembles in the logistic maps in one-dimensional case. We see that the topological entropy in the neighborhood of a = 1.4 is monotonically increase and there exist a few flat places; therefore, the dynamical system varies sensitively depending on the value of a.
3.2. Stable and unstable manifolds Fig. 3 shows the stable and unstable manifolds of the Hénon map. We see that the unstable manifold intersects the stable manifold almost everywhere. These manifolds are obtained from the asymptotic expansion x(t n) = f(M+1)[x(t)] in (4). The unstable manifold in this figure is depicted by all [x(tm), y(tm)] adopted for the calculation of the length LM (0 6 M 6 22): however, the attractor remains unchanged visually, even if we take points [x(tm), y(tm)] only for 18 6 M 6 22. For the stable manifold, f1 in (4) is given as f1 = logja1j 1.8582 for a = 1.4, b = 0.3, and a1 0.1560. The number of digits and ðNÞ the upper limit in (4) for the calculation of bN;N1 for the stable manifold are the same as those for the unstable manifold. Since the value of x(t n) with respect to real t is real for the stable manifold, we can depict the stable
Fig. 3. Stable and unstable manifolds of Hénon map described by asymptotic expansion given by (4), where a = 1.4 and b = 0.3, and the fixed point is shifted to the origin.
manifold (x,y) = [x(t n),bx(t n 1)] continuously with respect to real t. We take the region 19,988 6 t 6 20,000 for n = 20,000, i.e., the map iteration number (minus 1) M (the integer part of jt nj), where 0 6 M 6 11, for the stable manifold. Here, 105 grid points are considered. Since the stable manifold is not an attractor, it is more difficult to take a large iteration number M, in contrast to the unstable manifold. The stable manifold diverges in the first quadrant as N ? 1 in (4). We emphasize that M = 22, i.e., the 23th iteration of f (223th order polynomial) is realized for calculating the topological entropy.
4. Conclusion We have estimated the topological entropy of the Hénon attractor by using a function that describes the stable and unstable manifolds of the Hénon map. The topological entropy is estimated by calculating the length of curves in the attractor geometrically. Our estimation result supports the estimation made by Newhouse et al. [13].
Fig. 2. Partial topological entropy h12,M in the neighborhood of a = 1.4; (a) 1.31 6 a 6 1.41 and (b) 1.3292 6 a 6 1.41, where b = 0.3.
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
There also exists another function that can describe the stable and unstable manifolds of the Hénon map. This function, often called the Poincaré function, was an entire function and was first presented by Poincaré in 1890 [18]. The existence and regularity (smoothness) of the Poincaré function have been investigated in detail by Cabré et al. [19–21]. This function is given in the form of a Taylor series around a hyperbolic fixed point [21] and is applicable to depiction of the stable and unstable manifolds (although the detailed structure as found in Fig. 3 cannot be obtained, especially in the stable manifold). Because it is expressed in the form of the Taylor series, the Poincaré function well describes the manifolds near the fixed point; however, as pointed out by Newhouse et al. [13], the approximation by this function deteriorates fairly quickly when we consider points far from the fixed point. Therefore, we cannot calculate the topological entropy using this function. It is important to mention that the Poincaré function is different from the function presented in this paper. The derivation of topological entropy is important, because various physical quantities in thermodynamics of dissipative systems are expected to be derived from topological entropy. Our method is also applicable to systems in which the attractor does not exist or the invariant set is not hyperbolic, as long as the fixed point is hyperbolic. We have calculated here up to 300 digits (M = 22). However, the series (4) is an exact solution of Eq. (2) and mathematically, we can calculate the topological entropy with analytical accuracy by using this series. The function presented here can capture the exponentially small or large effects and that enables us to investigate various analytical structures of stable and unstable manifolds in addition to calculating topological entropy. We refer to the fact that the algorithm adopted in the present study cannot be applied to the case in which the fixed point is elliptic, i.e., the case of the eigenvalue jaj = 1. However, our algorithm is applicable to the entire range of parameters a and b except this case. Acknowledgements This work was partially supported by a Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science.
809
References [1] Hénon M. A two-dimensional mapping with a strange attractor. Commun Math Phys 1976;50:69–77. [2] Frisch U, Sulem PL, Nelkin M. A simple dynamical model of intermittent fully developed turbulence. J Fluid Mech 1978;87:719–36 [references therein]. [3] Mori H, Fujisaka H. Statistical dynamics of chaotic flows. Prog Theor Phys 1980;63:1931–44. [4] Eckmann JP, Ruelle D. Ergodic theory of chaos and strange attractors. Rev Mod Phys 1985;57:617–56 [references therein]. [5] Yomdin Y. Volume growth and entropy. Israel J Math 1987;57:285–300. [6] Newhouse S. Entropy and volume. Ergod Th Dyn Sys 1988;8:283–99. [7] Newhouse S, Pignataro T. On the estimation of topological entropy. J Stat Phys 1993;72:1331–51. [8] Biham O, Wenzel W. Characterization of unstable periodic orbits in chaotic attractors and repellers. Phys Rev Lett 1989;63:819–22. [9] Biham O, Wenzel W. Unstable periodic orbits and the symbolic dynamics of the complex Hénon map. Phys Rev A 1990;42:4639–46. [10] Galias Z, Zgliczyn´ski P. Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Héenon map. Nonlinearity 2001;14:909–32. [11] Galias Z. Interval methods for rigorous investigations of periodic orbits. Internat J Bifur Chaos 2001;9:2427–50. [12] Gelfert K, Kantz H. Dynamical quantities and their numerical analysis by saddle periodic orbits. Physica D 2007;232:166–72. [13] Newhouse S, Berz M, Grote J, Makino K. On the estimation of topological entropy on surfaces. Geometric and probabilistic structures in dynamics, Contemporary Math 2008;469:243–270 [Providence RI: Amer. Math. Soc.] [14] Bowen R. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math, vol. 470. Springer; 1975. [15] Arai Z. On hyperbolic plateaus of the Hénon maps. Experiment Math 2007;16:181–8. [16] Matsuoka C, Hiraide K. Special functions created by Borel–Laplace transform of Hénon map. Electro Res Announc Math Sci 2011;18:1–11. [17] Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études Sci Publ Math 1980;51:137–73. [18] Poincaré H. Sur une classe nouvelle de transcendantes uniformes. J de Math 1890;6:313–65. [19] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds I. Manifolds associated to non-resonant subspaces. Indiana Univ Math J 2003;52:283–328. [20] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds II. Regularity with respect to parameters. Indiana Univ Math J 2003;52:329–60. [21] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds III. Overview and applications. J Diff Eqs 2005;218:444–515.
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Entropy estimation of the Hénon attractor Chihiro Matsuoka a,⇑, Koichi Hiraide b a b
Department of Physics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japan Department of Mathematics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japan
a r t i c l e
i n f o
Article history: Received 8 June 2011 Accepted 18 February 2012 Available online 31 March 2012
a b s t r a c t The topological entropy of the Hénon attractor is estimated using a function that describes the stable and unstable manifolds of the Hénon map. This function provides an accurate estimate of the length of curves in the attractor. The estimation method presented here can be applied to cases in which the invariant set is not hyperbolic. From the result of the length calculation, we have estimated the topological entropy h as h 0.49703 for the original parameters a = 1.4 and b = 0.3 adopted by Hénon. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The Hénon map [1] is a model that depicts fluid turbulence from the viewpoint of a dynamical system [2]; in this map, the topological entropy of the attractor is the quantity that measures the complexity of fully developed turbulence [3,4]. Generally, when the topological entropy is known, we can use it to calculate thermodynamical quantities such as the Gibbs measure or the pressure of a system. Owing to the significance of entropy in thermodynamics of dissipative systems or the probability theory, several studies have calculated the topological entropy of the Hénon attractor by various methods. In the case of a two-dimensional system, the topological entropy can be defined as supremum of the growth ratio of lengths of curves in invariant sets (usually, the unstable manifold) in a system [5–7]. Mori and Fujisaka calculated the topological entropy h and the Housdorff dimension D of the Hénon map by the method of map iteration and obtained the values of h 0.42 and D 1.26. Biham and Wenzel introduced a Hamiltonian associated with the Hénon map and estimated as h 0.67 (note that their definition of topological entropy is slightly different from the one presented here) by counting the number of periodic points up to period p = 28 [8]. Their numerical technique can also be applied ⇑ Corresponding author. E-mail address: [email protected] (C. Matsuoka). 0960-0779/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2012.02.013
to depict Julia sets in complex dynamical systems [9]. Galias and Zgliczyn´ski [10] and Galias [11] calculated the topological entropy by counting periodic points up to period p = 30, in which they estimated the value h 0.46486. Gelfert and Kantz adopted the algorithm of Biham and Wenzel and use it to calculate the topological entropy and pressure for various values of dynamical parameters a and b [see (1)] by counting periodic points up to period p = 31, by which they estimated as h 0.44–0.46 for the Hénon map [12]. Newhouse et al. performed numerical calculations with rigorous error estimates and proved the result of h P 0.46469 [13]. When the invariant set is hyperbolic, the topological entropy is usually given by counting the number of periodic points [14]; however, it is very difficult to obtain an accurate value of topological entropy in cases where the invariant set is not hyperbolic or the unstable manifold is not a true attractor. Although a rigorous mathematical proof does not exist, the invariant set of the Hénon map is considered to be non-hyperbolic [15]; therefore, the method of calculating periodic points [8,12] does not always give an accurate entropy value. The method of counting parts of the stable and unstable manifolds created by the map iteration presented by Newhouse et al. [13] gives an accurate value; however, complicated numerical calculations are required for this method. Matsuoka and Hiraide found a function that describes the stable and unstable manifolds of the Hénon map and depicted these manifolds using this function [16]. This function provides the
806
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
accurate length of the unstable manifold; therefore, we can calculate the topological entropy by using this function regardless of whether or not the invariant set is hyperbolic, i.e., whether or not the unstable manifold is an true attractor. In this paper, we report estimation of the topological entropy of the Hénon map by adopting this function. We mention that this estimation method is applicable to various values of dynamical parameters a and b [see Definition (1)]. This paper is organized as follows. In Section 2, we present basic facts and notations adopted in this paper, and briefly review the asymptotic expansion form to describe the stable and unstable manifolds of the Hénon map. In Section 3, we provide the definition of the topological entropy and perform the numerical estimate of that using the asymptotic expansion presented in the previous section. A figure of the stable and unstable manifolds of the Hénon map represented by this asymptotic expansion is also presented in this section. Section 4 is devoted to conclusion. 2. Asymptotic expansion of stable and unstable manifolds of Hénon map The Hénon map [1] is an invertible nonlinear transformation defined by a polynomial map f : C2 ! C2
f ðx; yÞ ¼ ð1 þ y ax2 ; bxÞ;
ð1Þ
where a and b are complex parameters; and we set a = 1.4 and b = 0.3 when estimating the topological entropy. When a fixed point P = (xf, yf) is a saddle point, two eigenvalues a1 and a2 at P are given by the solutions to the quadratic equation a2 + 2axfa b = 0, where 0 < ja1j < 1 and ja2j > 1. We define the stable and unstable manifolds at P by
N log N N ðNÞ K N ja1 j bN;N1 6 N!
for the stable manifold and as N log N 0N ðNÞ K N ja2 j bN;N1 6 N!
for the unstable manifold, where KN > 0 and K 0N > 0 are constants. These estimates guarantee the suppression of divergence of the factor eNf1 t ðN 1Þ! for Re(f1t) > 0 in (3). Here, we mention that the asymptotic expansion (3) is not a formal one; that is, this expansion is not the divergent series (for details, see Ref. [16]). In order to depict the stable and unstable manifolds, we introduce a large positive integer n such that
xðt nÞ
1 P
ðNÞ
eNf1 ðtnÞ
N¼1
bN;N1 ðN 1Þ! ðtÞN
;
ð4Þ
where the upper (lower) sign in (±) corresponds to the stable (unstable) manifold. We denote the integer part of jRe(t) nj as M; then the integer M corresponds to the map iteration number (minus 1), i.e., f(M+1)[x(t)] = x(t n). When Re(t) > 0 and Re(t) n < 0, the curve (x, y) = [x(t n), bx(t n 1)] describes the stable manifold; when Re(t) > 0 and Re(t) n > 0, the curve [x(t n), bx(t n 1)] tends to the fixed point P owing to the relation limM?1fM[x(t)] ? P. When Re(t) < 0 and Re(t) + n > 0, the curve (x, y) = [x(t + n), bx(t + n 1)] describes the unstable manifold; when Re(t) < 0 and Re(t) + n < 0, the curve [x(t + n), bx(t + n 1)] tends to the fixed point P owing to the relation limM?1fM[x(t)] ? P. In order to calculate the length of the unstable manifold, we adopt the expansion (4). 3. Topological entropy of the Hénon attractor
W s ðPÞ ¼ fQ 2 C2 jf n ðQ Þ ! P as n ! 1g and
3.1. Definition and numerical estimate of topological entropy
W u ðPÞ ¼ fQ 2 C2 jf n ðQ Þ ! P as n ! 1g;
Following Yomdin, and Newhouse et al. [5–7], we adopt the topological entropy h as
respectively. After shifting the fixed point P to the origin by x ? x + xf and y ? y + yf, we introduce a parameter t 2 C that parameterizes the stable and unstable manifolds as f[x(t), y(t)] = [x(t + 1), y(t + 1)]. Then, the Hénon map (1) yields the difference equation
xðt þ 1Þ kxðtÞ bxðt 1Þ ¼ afxðtÞg2 ;
ð2Þ
associated with y(t) = bx(t 1), where k = 2axf. Using the Borel–Laplace transform, Matsuoka and Hiraide found solution x(t) to (2), whose asymptotic expansion is given as [16]
xðtÞ
1 P N¼1
ðNÞ
Nf1 t
e
bN;N1 ðN 1Þ! ðtÞN
;
ð3Þ
where the + and signs correspond to the stable and unstable manifolds, respectively, and f1 is given by f1 = logja1j for the stable manifold and f1 = logja2j + pi ðNÞ for the unstable manifold. The coefficient bN;N1 is a constant given by a recurrence formula (Theorem 2 in [16]) and is estimated as
X Lkþ1 1 M1 ln ; M!1 M Lk k¼0
h ¼ lim
ð5Þ
where Lk is the length of the kth piece (the length obtained by the (k + 1)th iteration of the map f) on the Hénon attractor. Here, the length L of a curve is approximately given by
L¼
K1 P
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½xðtmþ1 Þ xðt m Þ2 þ ½yðtmþ1 Þ yðt m Þ2 ;
ð6Þ
m¼0
where the set [x(tm), y(tm)] denotes the coordinate on the (x, y) plane that is parameterized by the parameter tm, i.e., the discretized t. If the difference Mtm tm+1 tm is sufficiently small, L in (6) can be expected to well approximate the true length of the curve Ltrue ¼ ffi R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdx=dtÞ2 þ ðdy=dtÞ2 dt. We adopt formula (6) for calculating the partial length Lk (k = 0, 1, 2, . . .) numerically. Here, we restrict ourselves to the curves in the unstable manifold.
807
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
Since we cannot take M ? 1 in (5) numerically, we define the partial entropy hl,M for numerical calculations as follows:
hl;M ¼
1 X 1 M Lkþ1 ln : M l k¼l Lk
ð7Þ
As is seen easily, h is given by h = liml?0 limM?1hl,M. We use the asymptotic expansion (4) to calculate the length of the Hénon attractor. The algorithm for deriving the length Lk in (7) is as follows. First, we calculate the ðNÞ coefficient bN;N1 in (4) by using the recurrence formula in Theorem 2 in Ref. [16]. In order to avoid a round-off erðNÞ ror, the calculation of bN;N1 is performed with 300 digits. The upper limit of the summation in (4) is taken as N = 600. Then, we calculate x(t + n) (the unstable case) in ðNÞ (4) by using this bN;N1 , where f1 = logja2j + pi 0.6543 + pi for a = 1.4, b = 0.3, and a2 1.9237. We set n = 20,000 here. For this n, we consider the region 20,000 6 Re(t) 6 19,978, so the map iteration number (minus 1) M; i.e., the integer part of jRe(t) + nj, is in the range 0 6 M 6 22. As can be seen from the value of f1, the value of x(t + n) with respect to real t is not real for the unstable manifold; therefore, we must seek points such that the imaginary part of (x, y) is close to zero on C2 in order to obtain real (x, y). To do this, we divide the complex t-plane into pieces and search the points at which the distance from the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi real axis satisfies the condition sðtÞ xi ðtÞ2 þ yi ðtÞ2 6 ð 1Þ, where xi(t) = Im[x(t + n)] and yi(t) = Im[bx (t + n 1)]. We select = 103 in this study. The points that satisfy this condition correspond to tm in (6). Upon setting t = tr + i ti, we found that tm appears on a line with slope ti/ tr 4.8 [16]. The sets [x(tm), y(tm)] obtained by this algorithm accurately depict the unstable manifold of the Hénon map (see Fig. 3). For a small map iteration number M, we can find points t = tm that satisfy s(t) 6 103 up to the neighborhood of the next integer M + 1, when the starting point of the detection is taken to be as M. For example, we can find such points approximately up to M + 0.9 for M = 0 and 1 (tr = 20,000 and 19,999, respectively), M + 0.8 for M = 2 and 3, and M + 0.7 for M = 4 and 5, starting from M. However, these points gradually decrease in number owing to the roundoff error, and finally, we few such points are found for M P 23. We find s(t) 6 103 points up to M + 0.1 for M = 22 (tr = 19,978); therefore, taking this value into account, we select the interval M 6 tr 6 M + 0.1 in tr (ti = 4.8tr) for all M. Dividing the interval 0.1 in one integer M into equally spaced grid points with a distance of Mtr (Mti = 4.8Mtr), we calculate the length of curves for all M (0 6 M 6 22). Here, we set Mtr = 1.0 104 for 0 6 M 6 7, Mtr = 5.0 105 for 8 6 M 6 17, and Mtr = 1.0 105 for 18 6 M 6 22; that is, we consider 1000, 5000, and 10,000 grid points for 0 6 M 6 7, 8 6 M 6 17, and 18 6 M 6 22, respectively. The calculations for x(t + n) = f(M+1)[(x(t)] are performed with 32, 64, 128, and 300 digits for 0 6 M 6 7, 8 6 M 6 17, 18 6 M 6 21, and M = 22, respectively (see Fig. 3). By using the above algorithm, we find [x(tm), y(tm)] that would provide the unstable manifold and calculate the
length LM (0 6 M 6 22) from formula (6). These lengths for a = 1.4 and b = 0.3 are listed in Table 1. The lengths LM for 0 6 M 6 11 are very small and the points [x(tm), y(tm)] for these M values stay in the neighborhood of the fixed point. Indeed, almost all parts of the Hénon attractor are depicted by the points [x(tm),y(tm)] for 12 6 M 6 22. The partial topological entropy hl,M in (7) for 0 6 M 6 11 varies widely and we cannot obtain an accurate value for it. Therefore, we do not adopt these LM (0 6 M 6 11) values for the topological entropy calculation. Fig. 1 shows the partial topological entropy h12,M, for which l = 12 in (7). After varying at relatively small M values, the partial topological entropy h12,M approaches a certain value. The h12,M values for 19 6 M 6 22 are estimated as h12,19 0.53874, h12,20 0.50556, h12,21 0.49853, and h12,22 0.49703. We regard this last h12,22 value as the asymptotic value of the topological entropy h in the approximation up to M = 22, i.e.,
h 0:49703:
ð8Þ
Our result supports the estimate of h P 0.46469 by Newhouse et al. [13]. We remark that Fig. 1 mathematically
Table 1 Length of curves in the unstable manifold. The first and third columns show the iteration number M, and the second and fourth columns show the partial length of the attractor, LM, at the iteration number M. M
LM
M
LM
0 1 2 3 4 5 6 7 8 9 10 11
7.1843 105 1.3821 104 2.6583 104 5.1151 104 9.8341 104 1.8938 103 3.6353 103 7.0211 103 1.3402 102 2.6162 102 4.8879 102 9.9124 102
12 13 14 15 16 17 18 19 20 21 22
0.17008 0.39014 0.44008 1.31912 1.65998 2.68724 4.77124 7.38664 9.70860 15.1093 24.5045
Fig. 1. Partial topological entropy h12,M versus map iteration number M, where the white circles show the calculation points of h12,M.
808
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
converges with M ? 1; however, the result up to M = 22 is best possible in actual calculations at present. The values obtained by Newhouse et al. and in the present study are larger than the value calculated by counting the periodic points [12]. Katok [17] proved a theorem that when the system is two-dimensional and the topological entropy h > 0, there exist hyperbolic subdynamical systems having entropy h infinitely close to h. This h is possible to calculate by counting the number of periodic points. The difference between the result presented by Gelfert and Kantz (h 0.44–0.46) and ours (h 0.49703) seems to suggest that there exist numerous periodic points with longer period in the Hénon attractor that cannot count within the period p = 31 considered in Ref. [12]. We show the topological entropy for various values of a in the neighborhood of a = 1.4 in Fig. 2, where the value of b is fixed at b = 0.3. The accuracy of the calculation is identical with that of Fig. 1. We find the devil’s staircase-like structure for the topological entropy, which resembles in the logistic maps in one-dimensional case. We see that the topological entropy in the neighborhood of a = 1.4 is monotonically increase and there exist a few flat places; therefore, the dynamical system varies sensitively depending on the value of a.
3.2. Stable and unstable manifolds Fig. 3 shows the stable and unstable manifolds of the Hénon map. We see that the unstable manifold intersects the stable manifold almost everywhere. These manifolds are obtained from the asymptotic expansion x(t n) = f(M+1)[x(t)] in (4). The unstable manifold in this figure is depicted by all [x(tm), y(tm)] adopted for the calculation of the length LM (0 6 M 6 22): however, the attractor remains unchanged visually, even if we take points [x(tm), y(tm)] only for 18 6 M 6 22. For the stable manifold, f1 in (4) is given as f1 = logja1j 1.8582 for a = 1.4, b = 0.3, and a1 0.1560. The number of digits and ðNÞ the upper limit in (4) for the calculation of bN;N1 for the stable manifold are the same as those for the unstable manifold. Since the value of x(t n) with respect to real t is real for the stable manifold, we can depict the stable
Fig. 3. Stable and unstable manifolds of Hénon map described by asymptotic expansion given by (4), where a = 1.4 and b = 0.3, and the fixed point is shifted to the origin.
manifold (x,y) = [x(t n),bx(t n 1)] continuously with respect to real t. We take the region 19,988 6 t 6 20,000 for n = 20,000, i.e., the map iteration number (minus 1) M (the integer part of jt nj), where 0 6 M 6 11, for the stable manifold. Here, 105 grid points are considered. Since the stable manifold is not an attractor, it is more difficult to take a large iteration number M, in contrast to the unstable manifold. The stable manifold diverges in the first quadrant as N ? 1 in (4). We emphasize that M = 22, i.e., the 23th iteration of f (223th order polynomial) is realized for calculating the topological entropy.
4. Conclusion We have estimated the topological entropy of the Hénon attractor by using a function that describes the stable and unstable manifolds of the Hénon map. The topological entropy is estimated by calculating the length of curves in the attractor geometrically. Our estimation result supports the estimation made by Newhouse et al. [13].
Fig. 2. Partial topological entropy h12,M in the neighborhood of a = 1.4; (a) 1.31 6 a 6 1.41 and (b) 1.3292 6 a 6 1.41, where b = 0.3.
C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809
There also exists another function that can describe the stable and unstable manifolds of the Hénon map. This function, often called the Poincaré function, was an entire function and was first presented by Poincaré in 1890 [18]. The existence and regularity (smoothness) of the Poincaré function have been investigated in detail by Cabré et al. [19–21]. This function is given in the form of a Taylor series around a hyperbolic fixed point [21] and is applicable to depiction of the stable and unstable manifolds (although the detailed structure as found in Fig. 3 cannot be obtained, especially in the stable manifold). Because it is expressed in the form of the Taylor series, the Poincaré function well describes the manifolds near the fixed point; however, as pointed out by Newhouse et al. [13], the approximation by this function deteriorates fairly quickly when we consider points far from the fixed point. Therefore, we cannot calculate the topological entropy using this function. It is important to mention that the Poincaré function is different from the function presented in this paper. The derivation of topological entropy is important, because various physical quantities in thermodynamics of dissipative systems are expected to be derived from topological entropy. Our method is also applicable to systems in which the attractor does not exist or the invariant set is not hyperbolic, as long as the fixed point is hyperbolic. We have calculated here up to 300 digits (M = 22). However, the series (4) is an exact solution of Eq. (2) and mathematically, we can calculate the topological entropy with analytical accuracy by using this series. The function presented here can capture the exponentially small or large effects and that enables us to investigate various analytical structures of stable and unstable manifolds in addition to calculating topological entropy. We refer to the fact that the algorithm adopted in the present study cannot be applied to the case in which the fixed point is elliptic, i.e., the case of the eigenvalue jaj = 1. However, our algorithm is applicable to the entire range of parameters a and b except this case. Acknowledgements This work was partially supported by a Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science.
809
References [1] Hénon M. A two-dimensional mapping with a strange attractor. Commun Math Phys 1976;50:69–77. [2] Frisch U, Sulem PL, Nelkin M. A simple dynamical model of intermittent fully developed turbulence. J Fluid Mech 1978;87:719–36 [references therein]. [3] Mori H, Fujisaka H. Statistical dynamics of chaotic flows. Prog Theor Phys 1980;63:1931–44. [4] Eckmann JP, Ruelle D. Ergodic theory of chaos and strange attractors. Rev Mod Phys 1985;57:617–56 [references therein]. [5] Yomdin Y. Volume growth and entropy. Israel J Math 1987;57:285–300. [6] Newhouse S. Entropy and volume. Ergod Th Dyn Sys 1988;8:283–99. [7] Newhouse S, Pignataro T. On the estimation of topological entropy. J Stat Phys 1993;72:1331–51. [8] Biham O, Wenzel W. Characterization of unstable periodic orbits in chaotic attractors and repellers. Phys Rev Lett 1989;63:819–22. [9] Biham O, Wenzel W. Unstable periodic orbits and the symbolic dynamics of the complex Hénon map. Phys Rev A 1990;42:4639–46. [10] Galias Z, Zgliczyn´ski P. Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Héenon map. Nonlinearity 2001;14:909–32. [11] Galias Z. Interval methods for rigorous investigations of periodic orbits. Internat J Bifur Chaos 2001;9:2427–50. [12] Gelfert K, Kantz H. Dynamical quantities and their numerical analysis by saddle periodic orbits. Physica D 2007;232:166–72. [13] Newhouse S, Berz M, Grote J, Makino K. On the estimation of topological entropy on surfaces. Geometric and probabilistic structures in dynamics, Contemporary Math 2008;469:243–270 [Providence RI: Amer. Math. Soc.] [14] Bowen R. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math, vol. 470. Springer; 1975. [15] Arai Z. On hyperbolic plateaus of the Hénon maps. Experiment Math 2007;16:181–8. [16] Matsuoka C, Hiraide K. Special functions created by Borel–Laplace transform of Hénon map. Electro Res Announc Math Sci 2011;18:1–11. [17] Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études Sci Publ Math 1980;51:137–73. [18] Poincaré H. Sur une classe nouvelle de transcendantes uniformes. J de Math 1890;6:313–65. [19] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds I. Manifolds associated to non-resonant subspaces. Indiana Univ Math J 2003;52:283–328. [20] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds II. Regularity with respect to parameters. Indiana Univ Math J 2003;52:329–60. [21] Cabré X, Fontich E, de la Llave R. The parameterization method for invariant manifolds III. Overview and applications. J Diff Eqs 2005;218:444–515.